Flip until heads, earn three dollars per flip

Question

You flip a fair coin over and over until the first time it lands heads, then you stop. You earn $3 for every flip you make, including the flip that finally lands heads.

On average, how much do you earn playing this game?

Accepted Answer

The payout is 3 for each flip, so first find the average number of flips, then multiply by 3. Average number of flips. There is a clean way to think about this. You always make at least one flip. That flip is heads half the time (you stop) and tails half the time (you keep going, and you are back in the exact same situation). Call the average number of flips N. You always spend one flip. Then, with probability one half, the coin was tails and you must play again, adding another N flips on average. In words, average flips equals one, plus a half-chance of starting over: N = 1 + 	frac{1}{2}N. …

Flip until heads, earn three dollars per flip

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